A few things: You wired up your circuit as common cathode display, your text says common anode. If it works, no problem, just change the label.Next, you have a current limiting resistors on both middle contacts. That's not what you want, they're connected internally and present on both sides for convenience only. Remove one.Last, you have the current limiting resistor on the common cathode. Which means, the current through led varies with how many segments are active, so the 8 will be much darker than the 1. It's better to put a resistor on each line from the Arduino to the anode of each led. The version with the resistor on the common cathode can also fry a led, if one draws more current than the others. But that shouln't be problem with a 7-segment led display, the leds are going to be pretty much equal.

Even I was confused whether it has a common cathode or common anode. The datasheet says it is common anode, but I am not very sure. Is there a way by which I can check whether it is common anode or cathode?